Permutations and Combinations

If one choice has m options and the next has n, together they have m × n outcomes. Everything else is built on this.

Factorials

n! multiplies every whole number from n down to 1. Five books arrange on a shelf in 5! = 120 ways.

Permutations

A permutation counts arrangements, where order matters. nPr = n! ÷ (n − r)!.

Combinations

A combination counts selections, where order does not matter. nCr = n! ÷ (r!(n − r)!).

The link

A combination is a permutation with the r! orderings divided out, so nCr is always smaller than nPr.

How to decide

Ask whether swapping two chosen items produces a different outcome. A president and a vice-president, yes. A committee of two, no.

Does order matter?

A permutation counts arrangements, where order matters. A combination counts selections, where it does not. Deciding which question you are answering is the whole difficulty; the formulas are then mechanical.

The two formulas

Permutations of r from n: n!/(n − r)!. Combinations: n!/(r!(n − r)!). The extra r! divides out the orderings you no longer wish to distinguish, which is exactly the difference between the two questions.

Factorials count arrangements

n! is the number of ways to order n distinct objects: n choices for first, n − 1 for second, and so on. And 0! = 1, because there is exactly one way to arrange nothing — a convention that keeps the formulas working.

A quick test

Would swapping two of your chosen items give a different outcome? A committee of Ana and Ben is the same as Ben and Ana, so it is a combination. First and second place is not, so it is a permutation.